The problem asks you to find the mean (expected value) of the given probability distribution. Let's compute it step by step:

The formula for the mean (expected value) of a probability distribution is:

$\mu = \sum_{i} x_i P(x_i)$

From the table in the image:

• $x = 0, P(x) = 0.15$
• $x = 1, P(x) = 0.10$
• $x = 2, P(x) = 0.11$
• $x = 3, P(x) = 0.65$

Step 1: Multiply each $x$ value by its corresponding $P(x)$:

• $0 \times 0.15 = 0.00$
• $1 \times 0.10 = 0.10$
• $2 \times 0.11 = 0.22$
• $3 \times 0.65 = 1.95$

Step 2: Add the results:

$0.00 + 0.10 + 0.22 + 1.95 = 2.27$

Step 3: Round to one decimal place:

The mean is $\mu = 2.3$.

Let me know if you'd like a detailed explanation or assistance with any other part!

Related Questions:

1. How do you find the standard deviation for this probability distribution?
2. What is the interpretation of the mean in this context?
3. How can you verify if this is a valid probability distribution?
4. How would the mean change if one $P(x)$ value increased or decreased?
5. What does it mean if the distribution is symmetric or skewed?

Tip:

Always ensure that the sum of all $P(x)$ values equals 1 to confirm the validity of a probability distribution.